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If X is a discrete random variable taking values on some subset of the non-negative integers, {0,1, ...}, then the probability-generating function of X is defined as:
Probability-generating functions obey all the rules of power series with non-negative coefficients. In particular, since G(1-) = 1 (since the probabilities must sum to one), the radius of convergence of any probability-generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients. (Note that G(1-) = limz↑1G(z).)
The following properties allow the derivation of various basic quantities related to X:
<math>\quad f(k) = \textrm{Pr}(X = k) = \frac{G^{(k)}(0)}{k!}.</math>
<math> \textrm{E}(X) = G'(1-).</math>
More generally, the kth factorial moment, E(X(X - 1) ... (X - k + 1)), of X is given by
<math>\textrm{E}\left(X(X-1)\ldots(X-k+1)\right) = G^{(k)}(1-), \quad k \geq 1.</math>
Probability-generating functions are particularly useful for dealing with sums of independent random variables. If X1, X2, ..., Xn is a sequence of independent (and not necessarily identically distributed) random variables, then if
<math>G(z) = z^c.</math>
<math>G(z) = \left[(1-p) + pz\right]^n.</math>
Note that this is the n-fold product of the probability-generating function of a Bernoulli random variable with parameter p.
<math>G(z) = \left(\frac{p}{1 - (1-p)z}\right)^r.</math>
Note that this is the r-fold product of the probabiltiy generating function of a geometric random variable.
<math>G(z) = \textrm{e}^{\lambda(z - 1)}.</math>
The probability-generating function is occasionally called the z-transform of the probability mass function. It is an example of a generating function of a sequence (see formal power series).
Other generating functions of random variables include the moment-generating function and the characteristic function.
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